Document eta-expansion workaround for Empty : Set

I thought I had pushed this before (see
https://github.com/HoTT/HoTT/pull/371#issuecomment-41201465).  But I
guess no.
Make lift and lower transparent

The reason to make them opaque was purely for typeclasses.  They will
need to be transparent when we move to trunk-polyproj, to get an
alternate version of Funext_downward_closed to work correctly.
Give a better error message on a tactic
Merge pull request #386 from JasonGross/transparent-lift-lower

Make lift and lower transparent
Add an instance about [idtoiso] and [components_of]
Add some typeclass resolution hints for apply being stupid

[apply] doesn't know that, e.g., the identity functor applied to x is
x, nor vice-versa.  And because it infers arguments in the "wrong"
order, it errors before it has a chance to pick up that information.
So we need to help it along a little.
Split out InitialTerminalCategory.v into multiple files

This way, we can add pseudofunctors without making the
initial/terminal categories depend on them.
Add pseudofunctors from initial/terminal categories

These will be useful for lax slice categories
Define the identity pseudofunctor

I'm not sure why it's so slow.  Maybe typeclass resolution on
[Reflexive]?

After    | File Name                                                          | Before   || Change
-----------------------------------------------------------------------------------------------------
9m46.05s | Total                                                              | 9m32.56s || +0m13.49s
-----------------------------------------------------------------------------------------------------
0m13.24s | categories/Pseudofunctor/Identity                                  |   N/A    || +0m13.24s
1m47.71s | categories/Comma/ProjectionFunctors                                | 1m48.81s || -0m01.09s
0m26.52s | categories/Grothendieck/PseudofunctorToCat                         | 0m25.30s || +0m01.21s
0m22.86s | categories/Adjoint/HomCoercions                                    | 0m20.94s || +0m01.91s
0m16.69s | categories/Adjoint/UnitCounitCoercions                             | 0m15.60s || +0m01.09s
0m15.43s | categories/PseudonaturalTransformation/Core                        | 0m14.18s || +0m01.25s
0m15.31s | categories/Comma/InducedFunctors                                   | 0m16.89s || -0m01.58s
0m32.82s | categories/Adjoint/Composition/AssociativityLaw                    | 0m33.05s || -0m00.22s
0m16.78s | categories/ExponentialLaws/Law4/Functors                           | 0m16.88s || -0m00.09s
0m15.28s | categories/FunctorCategory/Dual                                    | 0m15.74s || -0m00.46s
0m12.92s | categories/Functor/Pointwise/Properties                            | 0m12.05s || +0m00.86s
0m12.72s | hit/Flattening                                                     | 0m12.94s || -0m00.21s
0m12.14s | categories/Pseudofunctor/Core                                      | 0m11.96s || +0m00.17s
0m12.04s | categories/Adjoint/Composition/Core                                | 0m12.80s || -0m00.76s
0m11.45s | categories/Adjoint/Dual                                            | 0m12.40s || -0m00.95s
0m11.08s | categories/Adjoint/Composition/IdentityLaws                        | 0m10.38s || +0m00.69s
0m10.78s | categories/Pseudofunctor/FromFunctor                               | 0m11.07s || -0m00.29s
0m09.03s | categories/Yoneda                                                  | 0m08.56s || +0m00.46s
0m08.55s | categories/ExponentialLaws/Law1/Law                                | 0m08.65s || -0m00.09s
0m08.17s | categories/Functor/Prod/Universal                                  | 0m08.17s || +0m00.00s
0m07.58s | categories/GroupoidCategory/Morphisms                              | 0m08.05s || -0m00.47s
0m07.48s | categories/UniversalProperties                                     | 0m06.92s || +0m00.56s
0m07.08s | categories/Comma/Dual                                              | 0m06.14s || +0m00.94s
0m06.84s | hit/Connectedness                                                  | 0m07.50s || -0m00.66s
0m06.61s | categories/Structure/IdentityPrinciple                             | 0m06.82s || -0m00.20s
0m06.42s | categories/Adjoint/Paths                                           | 0m06.66s || -0m00.24s
0m05.80s | categories/Category/Morphisms                                      | 0m05.55s || +0m00.25s
0m05.78s | categories/Adjoint/UnitCounit                                      | 0m06.27s || -0m00.48s
0m05.68s | categories/Functor/Paths                                           | 0m05.90s || -0m00.22s
0m05.57s | categories/Functor/Sum                                             | 0m05.54s || +0m00.03s
0m05.32s | categories/DualFunctor                                             | 0m05.02s || +0m00.30s
0m05.20s | EquivalenceVarieties                                               | 0m05.09s || +0m00.11s
0m04.70s | categories/ExponentialLaws/Law2/Functors                           | 0m04.90s || -0m00.20s
0m04.43s | categories/Category/Sigma/OnMorphisms                              | 0m04.58s || -0m00.15s
0m03.74s | categories/ProductLaws                                             | 0m03.78s || -0m00.03s
0m03.54s | categories/Structure/Core                                          | 0m03.56s || -0m00.02s
0m03.42s | types/ObjectClassifier                                             | 0m03.42s || +0m00.00s
0m02.99s | types/Sigma                                                        | 0m02.97s || +0m00.02s
0m02.91s | categories/Adjoint/UniversalMorphisms                              | 0m02.90s || +0m00.01s
0m02.90s | categories/Functor/Composition/Functorial/Attributes               | 0m03.07s || -0m00.16s
0m02.85s | HSet                                                               | 0m02.84s || +0m00.01s
0m02.72s | categories/Functor/Prod/Functorial                                 | 0m02.95s || -0m00.23s
0m02.72s | categories/FunctorCategory/Functorial                              | 0m02.75s || -0m00.02s
0m02.70s | HProp                                                              | 0m02.75s || -0m00.04s
0m02.63s | categories/Grothendieck/ToSet                                      | 0m02.64s || -0m00.01s
0m02.48s | Equivalences                                                       | 0m02.56s || -0m00.08s
0m02.29s | categories/Functor/Attributes                                      | 0m02.29s || +0m00.00s
0m02.25s | categories/Category/Sigma/Core                                     | 0m02.03s || +0m00.22s
0m02.22s | categories/Comma/Core                                              | 0m01.98s || +0m00.24s
0m02.18s | categories/NaturalTransformation/Isomorphisms                      | 0m02.07s || +0m00.11s
0m02.11s | categories/Category/Sum                                            | 0m02.35s || -0m00.24s
0m02.07s | types/Record                                                       | 0m02.02s || +0m00.04s
0m01.91s | PathGroupoids                                                      | 0m01.90s || +0m00.01s
0m01.88s | categories/NaturalTransformation/Dual                              | 0m01.86s || +0m00.01s
0m01.80s | categories/FunctorCategory/Morphisms                               | 0m01.75s || +0m00.05s
0m01.76s | categories/NaturalTransformation/Paths                             | 0m01.75s || +0m00.01s
0m01.75s | categories/HomFunctor                                              | 0m01.75s || +0m00.00s
0m01.31s | categories/Category/Sigma/OnObjects                                | 0m01.24s || +0m00.07s
0m01.29s | categories/Category/Prod                                           | 0m01.29s || +0m00.00s
0m01.28s | categories/InitialTerminalCategory/Pseudofunctors                  | 0m01.41s || -0m00.12s
0m01.23s | types/Prod                                                         | 0m01.31s || -0m00.08s
0m01.18s | categories/Limits/Core                                             | 0m01.14s || +0m00.04s
0m01.17s | categories/NaturalTransformation/Pointwise                         | 0m01.33s || -0m00.16s
0m01.15s | categories/NaturalTransformation/Composition/Laws                  | 0m01.13s || +0m00.02s
0m01.15s | categories/ExponentialLaws/Law1/Functors                           | 0m01.09s || +0m00.05s
0m01.06s | categories/Functor/Pointwise                                       | 0m01.10s || -0m00.04s
0m01.05s | categories/Functor/Composition/Functorial/Core                     | 0m01.21s || -0m00.15s
0m01.03s | categories/ExponentialLaws/Law  N/A                                | 0m01.10s || -0m00.07s
0m01.00s | categories/CategoryOfSections/Core                                 | 0m00.98s || +0m00.02s
0m00.96s | categories/Functor/Prod                                            | 0m00.86s || +0m00.09s
0m00.93s | contrib/HoTTBookExercises                                          | 0m00.92s || +0m00.01s
0m00.90s | hit/epi                                                            | 0m00.93s || -0m00.03s
0m00.88s | categories/NaturalTransformation/Prod                              | 0m00.89s || -0m00.01s
0m00.88s | types/Paths                                                        | 0m00.88s || +0m00.00s
0m00.88s | hit/Spheres                                                        | 0m01.12s || -0m00.24s
0m00.81s | categories/SetCategory/Morphisms                                   | 0m00.82s || -0m00.00s
0m00.79s | contrib/HoTTBook                                                   | 0m00.74s || +0m00.05s
0m00.67s | categories/categories                                              | 0m00.60s || +0m00.07s
0m00.64s | Trunc                                                              | 0m00.60s || +0m00.04s
0m00.63s | types/Sum                                                          | 0m00.62s || +0m00.01s
0m00.63s | types/Universe                                                     | 0m00.67s || -0m00.04s
0m00.62s | UnivalenceImpliesFunext                                            | 0m00.63s || -0m00.01s
0m00.62s | categories/NaturalTransformation/Composition/Core                  | 0m00.65s || -0m00.03s
0m00.61s | Functorish                                                         | 0m00.64s || -0m00.03s
0m00.58s | hit/Suspension                                                     | 0m00.71s || -0m00.13s
0m00.56s | categories/Comma/Projection                                        | 0m00.55s || +0m00.01s
0m00.55s | hit/quotient                                                       | 0m00.54s || +0m00.01s
0m00.52s | categories/Limits/Functors                                         | 0m00.54s || -0m00.02s
0m00.52s | categories/NaturalTransformation/Sum                               | 0m00.55s || -0m00.03s
0m00.51s | categories/Functor/Composition/Laws                                | 0m00.54s || -0m00.03s
0m00.50s | categories/ExponentialLaws/Law3/Functors                           | 0m00.46s || +0m00.03s
0m00.48s | hit/Circle                                                         | 0m00.49s || -0m00.01s
0m00.46s | categories/KanExtensions/Functors                                  | 0m00.52s || -0m00.06s
0m00.44s | categories/FundamentalPreGroupoidCategory                          | 0m00.34s || +0m00.09s
0m00.44s | categories/InitialTerminalCategory/Functors                        | 0m00.44s || +0m00.00s
0m00.44s | types/Forall                                                       | 0m00.45s || -0m00.01s
0m00.42s | Tactics                                                            | 0m00.40s || +0m00.01s
0m00.41s | categories/Cat/Core                                                | 0m00.38s || +0m00.02s
0m00.40s | categories/NaturalTransformation/Identity                          | 0m00.44s || -0m00.03s
0m00.36s | categories/HomotopyPreCategory                                     | 0m00.40s || -0m00.04s
0m00.36s | Fibrations                                                         | 0m00.34s || +0m00.01s
0m00.35s | TruncType                                                          | 0m00.32s || +0m00.02s
0m00.34s | categories/InitialTerminalCategory/NaturalTransformations          | 0m00.40s || -0m00.06s
0m00.33s | categories/Functor/Dual                                            | 0m00.31s || +0m00.02s
0m00.33s | categories/Functor/Functor                                         | 0m00.31s || +0m00.02s
0m00.32s | categories/IndiscreteCategory                                      | 0m00.31s || +0m00.01s
0m00.32s | categories/NaturalTransformation/Composition/Functorial            | 0m00.26s || +0m00.06s
0m00.31s | categories/KanExtensions/Core                                      | 0m00.31s || +0m00.00s
0m00.29s | categories/Adjoint/Adjoint                                         | 0m00.22s || +0m00.06s
0m00.28s | categories/Utf8                                                    | 0m00.28s || +0m00.00s
0m00.28s | categories/Cat/Morphisms                                           | 0m00.31s || -0m00.02s
0m00.28s | categories/Grothendieck/Grothendieck                               | 0m00.31s || -0m00.02s
0m00.26s | categories/ExponentialLaws/ExponentialLaws                         | 0m00.27s || -0m00.01s
0m00.26s | hit/minus1Trunc                                                    | 0m00.29s || -0m00.02s
0m00.26s | types/Arrow                                                        | 0m00.30s || -0m00.03s
0m00.25s | categories/SetCategory/Functors/Functors                           | 0m00.21s || +0m00.04s
0m00.24s | categories/Category/Objects                                        | 0m00.24s || +0m00.00s
0m00.24s | categories/Functor/Composition/Core                                | 0m00.23s || +0m00.00s
0m00.24s | categories/Notations                                               | 0m00.27s || -0m00.03s
0m00.23s | categories/NaturalTransformation/NaturalTransformation             | 0m00.22s || +0m00.01s
0m00.23s | categories/Adjoint/Hom                                             | 0m00.31s || -0m00.07s
0m00.22s | categories/Limits/Limits                                           | 0m00.21s || +0m00.01s
0m00.22s | categories/Grothendieck/ToCat                                      | 0m00.20s || +0m00.01s
0m00.21s | categories/Functor/Composition/Functorial                          | 0m00.24s || -0m00.03s
0m00.21s | categories/Functor/Utf8                                            | 0m00.20s || +0m00.00s
0m00.21s | categories/FunctorCategory/Core                                    | 0m00.16s || +0m00.04s
0m00.20s | categories/DependentProduct                                        | 0m00.23s || -0m00.03s
0m00.20s | categories/KanExtensions/KanExtensions                             | 0m00.18s || +0m00.02s
0m00.20s | categories/Functor/Notations                                       | 0m00.18s || +0m00.02s
0m00.20s | Modality                                                           | 0m00.21s || -0m00.00s
0m00.20s | categories/Comma/Notations                                         | 0m00.14s || +0m00.06s
0m00.20s | categories/PseudonaturalTransformation/PseudonaturalTransformation | 0m00.17s || +0m00.03s
0m00.20s | FunextVarieties                                                    | 0m00.18s || +0m00.02s
0m00.20s | types/Bool                                                         | 0m00.19s || +0m00.01s
0m00.20s | categories/InitialTerminalCategory/InitialTerminalCategory         | 0m00.26s || -0m00.06s
0m00.20s | categories/NaturalTransformation/Utf8                              | 0m00.20s || +0m00.00s
0m00.19s | hit/iso                                                            | 0m00.14s || +0m00.04s
0m00.19s | categories/Functor/Composition/Composition                         | 0m00.20s || -0m00.01s
0m00.19s | categories/Adjoint/Identity                                        | 0m00.17s || +0m00.01s
0m00.19s | categories/Comma/Comma                                             | 0m00.20s || -0m00.01s
0m00.19s | categories/NatCategory                                             | 0m00.19s || +0m00.00s
0m00.18s | categories/FunctorCategory/Utf8                                    | 0m00.22s || -0m00.04s
0m00.18s | categories/Functor/Prod/Prod                                       | 0m00.17s || +0m00.00s
0m00.18s | categories/FunctorCategory/FunctorCategory                         | 0m00.22s || -0m00.04s
0m00.18s | categories/GroupoidCategory/Core                                   | 0m00.16s || +0m00.01s
0m00.18s | categories/ExponentialLaws/Law2/Law2                               | 0m00.14s || +0m00.03s
0m00.18s | Overture                                                           | 0m00.16s || +0m00.01s
0m00.17s | Conjugation                                                        | 0m00.22s || -0m00.04s
0m00.17s | categories/Comma/Utf8                                              | 0m00.18s || -0m00.00s
0m00.17s | categories/ExponentialLaws/Law3/Law3                               | 0m00.17s || +0m00.00s
0m00.17s | categories/Adjoint/Utf8                                            | 0m00.13s || +0m00.04s
0m00.17s | categories/InitialTerminalCategory/Notations                       | 0m00.20s || -0m00.03s
0m00.16s | categories/Cat/Cat                                                 | 0m00.16s || +0m00.00s
0m00.16s | HoTT                                                               | 0m00.18s || -0m00.01s
0m00.16s | categories/NaturalTransformation/Notations                         | 0m00.16s || +0m00.00s
0m00.15s | categories/Category/Dual                                           | 0m00.19s || -0m00.04s
0m00.15s | Misc                                                               | 0m00.12s || +0m00.03s
0m00.15s | categories/Adjoint/Notations                                       | 0m00.16s || -0m00.01s
0m00.15s | categories/Adjoint/Composition/Composition                         | 0m00.16s || -0m00.01s
0m00.15s | Tests                                                              | 0m00.16s || -0m00.01s
0m00.15s | categories/ExponentialLaws/Law1/Law1                               | 0m00.20s || -0m00.05s
0m00.15s | categories/Structure/Utf8                                          | 0m00.07s || +0m00.07s
0m00.15s | categories/Category/Pi                                             | 0m00.15s || +0m00.00s
0m00.14s | categories/Category/Utf8                                           | 0m00.13s || +0m00.01s
0m00.14s | Contractible                                                       | 0m00.09s || +0m00.05s
0m00.14s | hit/Interval                                                       | 0m00.17s || -0m00.03s
0m00.14s | categories/InitialTerminalCategory/Core                            | 0m00.17s || -0m00.03s
0m00.14s | categories/Category/Univalent                                      | 0m00.13s || +0m00.01s
0m00.14s | categories/Profunctor/Profunctor                                   | 0m00.15s || -0m00.00s
0m00.14s | categories/Category/Category                                       | 0m00.18s || -0m00.03s
0m00.14s | categories/NaturalTransformation/Core                              | 0m00.16s || -0m00.01s
0m00.14s | categories/Pseudofunctor/Pseudofunctor                             | 0m00.16s || -0m00.01s
0m00.14s | categories/Category/Core                                           | 0m00.16s || -0m00.01s
0m00.14s | categories/CategoryOfSections/CategoryOfSections                   | 0m00.16s || -0m00.01s
0m00.13s | categories/Structure/Structure                                     | 0m00.11s || +0m00.02s
0m00.13s | coq/Init/Peano                                                     | 0m00.09s || +0m00.04s
0m00.13s | categories/ExponentialLaws/Law4/Law4                               | 0m00.12s || +0m00.01s
0m00.13s | hit/unique_choice                                                  | 0m00.16s || -0m00.03s
0m00.13s | categories/NaturalTransformation/Composition/Composition           | 0m00.15s || -0m00.01s
0m00.13s | types/Unit                                                         | 0m00.11s || +0m00.02s
0m00.12s | categories/Profunctor/Representable                                | 0m00.14s || -0m00.02s
0m00.12s | categories/Profunctor/Core                                         | 0m00.12s || +0m00.00s
0m00.12s | categories/SetCategory/Core                                        | 0m00.16s || -0m00.04s
0m00.12s | hit/TwoSphere                                                      | 0m00.12s || +0m00.00s
0m00.12s | categories/SetCategory/SetCategory                                 | 0m00.11s || +0m00.00s
0m00.12s | categories/Category/Notations                                      | 0m00.12s || +0m00.00s
0m00.12s | categories/Profunctor/Identity                                     | 0m00.14s || -0m00.02s
0m00.12s | UnivalenceAxiom                                                    | 0m00.10s || +0m00.01s
0m00.12s | hit/Truncations                                                    | 0m00.15s || -0m00.03s
0m00.12s | categories/Adjoint/Composition/LawsTactic                          | 0m00.14s || -0m00.02s
0m00.12s | categories/Profunctor/Utf8                                         | 0m00.14s || -0m00.02s
0m00.10s | categories/DiscreteCategory                                        | 0m00.11s || -0m00.00s
0m00.10s | categories/Adjoint/Core                                            | 0m00.13s || -0m00.03s
0m00.10s | categories/Profunctor/Notations                                    | 0m00.13s || -0m00.03s
0m00.10s | categories/Functor/Core                                            | 0m00.12s || -0m00.01s
0m00.10s | categories/Category/Subcategory/Wide                               | 0m00.10s || +0m00.00s
0m00.10s | categories/Structure/Notations                                     | 0m00.11s || -0m00.00s
0m00.10s | types/UniverseLevel                                                | 0m00.08s || +0m00.02s
0m00.10s | categories/Category/Subcategory/Subcategory                        | 0m00.12s || -0m00.01s
0m00.10s | categories/Category/Sigma/Sigma                                    | 0m00.12s || -0m00.01s
0m00.10s | categories/Functor/Pointwise/Pointwise                             | 0m00.14s || -0m00.04s
0m00.09s | categories/GroupoidCategory/GroupoidCategory                       | 0m00.12s || -0m00.03s
0m00.09s | categories/Functor/Identity                                        | 0m00.10s || -0m00.01s
0m00.09s | categories/FunctorCategory/Notations                               | 0m00.10s || -0m00.01s
0m00.08s | categories/Category/Strict                                         | 0m00.07s || +0m00.00s
0m00.08s | categories/Category/Subcategory/Full                               | 0m00.12s || -0m00.03s
0m00.07s | coq/Init/Specif                                                    | 0m00.06s || +0m00.01s
0m00.07s | Pointed                                                            | 0m00.06s || +0m00.01s
0m00.07s | coq/Program/Tactics                                                | 0m00.04s || +0m00.03s
0m00.06s | coq/Init/Datatypes                                                 | 0m00.08s || -0m00.02s
0m00.06s | coq/Init/Logic_Type                                                | 0m00.06s || +0m00.00s
0m00.05s | FunextAxiom                                                        | 0m00.07s || -0m00.02s
0m00.05s | types/Empty                                                        | 0m00.09s || -0m00.03s
0m00.04s | coq/Init/Logic                                                     | 0m00.02s || +0m00.02s
0m00.04s | coq/Init/Prelude                                                   | 0m00.02s || +0m00.02s
0m00.03s | coq/Init/Notations                                                 | 0m00.04s || -0m00.01s
0m00.03s | coq/Init/Tactics                                                   | 0m00.03s || +0m00.00s
0m00.02s | coq/Bool/Bool                                                      | 0m00.04s || -0m00.02s