Port Matthieu's code formalizing pushouts

This code will allow the epi -> surj in set_cat proof to go through without
hacks and without universe inconsistencies in trunk.

I'm not entirely happy with the proofs.

This closes #402 and #404.

After    | File Name                                                          | Before   || Change
-----------------------------------------------------------------------------------------------------
6m16.18s | Total                                                              | 6m10.64s || +0m05.54s
-----------------------------------------------------------------------------------------------------
0m06.31s | categories/Adjoint/Composition/IdentityLaws                        | 0m03.89s || +0m02.41s
0m02.62s | hit/epi                                                            | 0m00.44s || +0m02.18s
1m04.82s | categories/Pseudofunctor/RewriteLaws                               | 1m03.77s || +0m01.04s
0m21.56s | categories/Grothendieck/PseudofunctorToCat                         | 0m19.70s || +0m01.85s
0m14.26s | categories/Comma/InducedFunctors                                   | 0m13.02s || +0m01.24s
0m11.20s | categories/PseudonaturalTransformation/Core                        | 0m12.48s || -0m01.28s
0m08.89s | categories/Adjoint/HomCoercions                                    | 0m07.85s || +0m01.04s
0m39.22s | categories/Comma/ProjectionFunctors                                | 0m40.10s || -0m00.88s
0m17.15s | categories/ExponentialLaws/Law4/Functors                           | 0m17.40s || -0m00.25s
0m11.39s | categories/Adjoint/Composition/AssociativityLaw                    | 0m11.20s || +0m00.19s
0m11.04s | categories/Adjoint/Dual                                            | 0m11.66s || -0m00.62s
0m09.20s | categories/FunctorCategory/Dual                                    | 0m09.02s || +0m00.17s
0m08.66s | categories/Yoneda                                                  | 0m09.50s || -0m00.83s
0m07.66s | categories/Adjoint/UnitCounit                                      | 0m07.72s || -0m00.05s
0m05.15s | categories/UniversalProperties                                     | 0m05.78s || -0m00.62s
0m05.09s | categories/Adjoint/UnitCounitCoercions                             | 0m04.84s || +0m00.25s
0m05.02s | categories/Adjoint/UniversalMorphisms                              | 0m04.87s || +0m00.14s
0m04.57s | categories/Pseudofunctor/Core                                      | 0m04.46s || +0m00.11s
0m04.23s | categories/Pseudofunctor/FromFunctor                               | 0m04.23s || +0m00.00s
0m04.23s | categories/Pseudofunctor/Identity                                  | 0m04.20s || +0m00.03s
0m04.15s | categories/Comma/Dual                                              | 0m04.66s || -0m00.50s
0m03.46s | categories/Limits/Core                                             | 0m02.96s || +0m00.50s
0m03.16s | categories/Structure/IdentityPrinciple                             | 0m03.18s || -0m00.02s
0m03.16s | categories/Functor/Pointwise/Properties                            | 0m03.20s || -0m00.04s
0m03.15s | categories/GroupoidCategory/Morphisms                              | 0m03.30s || -0m00.14s
0m03.10s | categories/Adjoint/Composition/Core                                | 0m03.98s || -0m00.87s
0m03.04s | categories/Category/Morphisms                                      | 0m03.04s || +0m00.00s
0m02.54s | categories/Functor/Composition/Functorial/Attributes               | 0m02.46s || +0m00.08s
0m02.42s | categories/Functor/Composition/Functorial/Core                     | 0m02.53s || -0m00.10s
0m02.41s | hit/Flattening                                                     | 0m02.60s || -0m00.18s
0m02.37s | categories/ExponentialLaws/Law1/Law                                | 0m02.59s || -0m00.21s
0m02.30s | categories/Functor/Prod/Functorial                                 | 0m02.24s || +0m00.05s
0m01.98s | categories/Functor/Attributes                                      | 0m01.68s || +0m00.30s
0m01.97s | categories/ExponentialLaws/Law2/Functors                           | 0m02.12s || -0m00.15s
0m01.85s | categories/ProductLaws                                             | 0m01.86s || -0m00.01s
0m01.81s | categories/Category/Sigma/OnMorphisms                              | 0m01.90s || -0m00.08s
0m01.76s | categories/NaturalTransformation/Isomorphisms                      | 0m01.79s || -0m00.03s
0m01.75s | hit/Connectedness                                                  | 0m01.64s || +0m00.11s
0m01.72s | categories/ExponentialLaws/Law  N/A                                | 0m01.46s || +0m00.26s
0m01.72s | categories/Functor/Prod/Universal                                  | 0m01.77s || -0m00.05s
0m01.62s | categories/Functor/Sum                                             | 0m01.61s || +0m00.01s
0m01.58s | categories/NaturalTransformation/Pointwise                         | 0m01.62s || -0m00.04s
0m01.58s | categories/InitialTerminalCategory/Pseudofunctors                  | 0m01.47s || +0m00.11s
0m01.50s | categories/ExponentialLaws/Law1/Functors                           | 0m01.07s || +0m00.42s
0m01.49s | categories/Functor/Paths                                           | 0m01.54s || -0m00.05s
0m01.43s | categories/Functor/Pointwise                                       | 0m01.54s || -0m00.11s
0m01.42s | categories/Limits/Functors                                         | 0m01.42s || +0m00.00s
0m01.36s | categories/FunctorCategory/Functorial                              | 0m01.23s || +0m00.13s
0m01.23s | categories/FunctorCategory/Morphisms                               | 0m01.16s || +0m00.07s
0m01.20s | categories/Category/Sum                                            | 0m01.17s || +0m00.03s
0m01.19s | categories/Adjoint/Paths                                           | 0m01.56s || -0m00.37s
0m01.18s | Equivalences                                                       | 0m01.28s || -0m00.10s
0m01.15s | HSet                                                               | 0m01.10s || +0m00.04s
0m01.14s | HProp                                                              | 0m01.12s || +0m00.01s
0m01.12s | types/Sigma                                                        | 0m01.14s || -0m00.01s
0m01.12s | categories/DualFunctor                                             | 0m01.36s || -0m00.24s
0m01.07s | PathGroupoids                                                      | 0m01.04s || +0m00.03s
0m00.99s | categories/KanExtensions/Functors                                  | 0m00.96s || +0m00.03s
0m00.95s | categories/Functor/Prod                                            | 0m00.87s || +0m00.07s
0m00.94s | types/Record                                                       | 0m01.04s || -0m00.10s
0m00.94s | categories/Structure/Core                                          | 0m00.82s || +0m00.12s
0m00.88s | EquivalenceVarieties                                               | 0m00.85s || +0m00.03s
0m00.86s | categories/Category/Sigma/Core                                     | 0m00.94s || -0m00.07s
0m00.82s | categories/Comma/Core                                              | 0m00.92s || -0m00.10s
0m00.80s | categories/NaturalTransformation/Prod                              | 0m00.84s || -0m00.03s
0m00.76s | categories/HomFunctor                                              | 0m00.75s || +0m00.01s
0m00.75s | categories/Grothendieck/ToCat                                      | 0m00.77s || -0m00.02s
0m00.74s | types/ObjectClassifier                                             | 0m00.71s || +0m00.03s
0m00.72s | contrib/HoTTBookExercises                                          | 0m00.60s || +0m00.12s
0m00.69s | categories/Category/Prod                                           | 0m00.66s || +0m00.02s
0m00.68s | categories/Category/Sigma/OnObjects                                | 0m00.68s || +0m00.00s
0m00.62s | hit/Spheres                                                        | 0m00.57s || +0m00.05s
0m00.59s | categories/NaturalTransformation/Dual                              | 0m00.61s || -0m00.02s
0m00.58s | categories/Grothendieck/ToSet                                      | 0m00.65s || -0m00.07s
0m00.57s | categories/KanExtensions/Core                                      | 0m00.56s || +0m00.00s
0m00.54s | types/Prod                                                         | 0m00.55s || -0m00.01s
0m00.54s | types/Paths                                                        | 0m00.50s || +0m00.04s
0m00.52s | categories/InitialTerminalCategory/Functors                        | 0m00.44s || +0m00.08s
0m00.51s | categories/SetCategory/Morphisms                                   | 0m00.47s || +0m00.04s
0m00.48s | categories/categories                                              | 0m00.53s || -0m00.05s
0m00.48s | categories/Comma/Projection                                        | 0m00.47s || +0m00.01s
0m00.46s | categories/ExponentialLaws/Law3/Functors                           | 0m00.50s || -0m00.03s
0m00.46s | categories/NaturalTransformation/Composition/Laws                  | 0m00.44s || +0m00.02s
0m00.45s | contrib/HoTTBook                                                   | 0m00.54s || -0m00.09s
0m00.43s | hit/Circle                                                         | 0m00.43s || +0m00.00s
0m00.41s | types/Universe                                                     | 0m00.39s || +0m00.01s
0m00.39s | categories/NaturalTransformation/Paths                             | 0m00.41s || -0m00.01s
0m00.39s | categories/CategoryOfSections/Core                                 | 0m00.38s || +0m00.01s
0m00.36s | categories/NaturalTransformation/Composition/Functorial            | 0m00.35s || +0m00.01s
0m00.35s | hit/quotient                                                       | 0m00.34s || +0m00.00s
0m00.34s | types/Sum                                                          | 0m00.32s || +0m00.02s
0m00.33s | Tactics                                                            | 0m00.27s || +0m00.06s
0m00.30s | hit/Suspension                                                     | 0m00.30s || +0m00.00s
0m00.30s | types/Forall                                                       | 0m00.27s || +0m00.02s
0m00.29s | categories/NaturalTransformation/Sum                               | 0m00.26s || +0m00.02s
0m00.29s | categories/Cat/Morphisms                                           | 0m00.21s || +0m00.07s
0m00.28s | categories/DependentProduct                                        | 0m00.31s || -0m00.02s
0m00.28s | categories/Cat/Core                                                | 0m00.34s || -0m00.06s
0m00.28s | categories/Functor/Composition/Laws                                | 0m00.25s || +0m00.03s
0m00.28s | Fibrations                                                         | 0m00.27s || +0m00.01s
0m00.28s | categories/NaturalTransformation/Composition/Core                  | 0m00.29s || -0m00.00s
0m00.27s | categories/Functor/Composition/Composition                         | 0m00.16s || +0m00.11s
0m00.27s | hit/Pushout                                                        |   N/A    || +0m00.27s
0m00.26s | Functorish                                                         | 0m00.22s || +0m00.04s
0m00.26s | categories/Functor/Functor                                         | 0m00.25s || +0m00.01s
0m00.25s | categories/Functor/Dual                                            | 0m00.22s || +0m00.03s
0m00.25s | categories/Adjoint/Hom                                             | 0m00.31s || -0m00.06s
0m00.24s | categories/ExponentialLaws/ExponentialLaws                         | 0m00.24s || +0m00.00s
0m00.24s | Trunc                                                              | 0m00.27s || -0m00.03s
0m00.24s | categories/Adjoint/Adjoint                                         | 0m00.21s || +0m00.03s
0m00.23s | HoTT                                                               | 0m00.22s || +0m00.01s
0m00.23s | categories/Functor/Notations                                       | 0m00.20s || +0m00.03s
0m00.23s | Tests                                                              | 0m00.21s || +0m00.02s
0m00.22s | categories/FundamentalPreGroupoidCategory                          | 0m00.25s || -0m00.03s
0m00.22s | UnivalenceImpliesFunext                                            | 0m00.29s || -0m00.06s
0m00.22s | categories/IndiscreteCategory                                      | 0m00.27s || -0m00.05s
0m00.21s | categories/InitialTerminalCategory/NaturalTransformations          | 0m00.23s || -0m00.02s
0m00.21s | categories/Functor/Composition/Functorial                          | 0m00.17s || +0m00.03s
0m00.21s | hit/minus1Trunc                                                    | 0m00.24s || -0m00.03s
0m00.21s | types/Arrow                                                        | 0m00.18s || +0m00.03s
0m00.21s | categories/NaturalTransformation/Identity                          | 0m00.16s || +0m00.04s
0m00.20s | categories/ExponentialLaws/Law3/Law3                               | 0m00.16s || +0m00.04s
0m00.20s | categories/Adjoint/Identity                                        | 0m00.19s || +0m00.01s
0m00.20s | TruncType                                                          | 0m00.22s || -0m00.01s
0m00.20s | categories/InitialTerminalCategory/InitialTerminalCategory         | 0m00.18s || +0m00.02s
0m00.20s | categories/Grothendieck/Grothendieck                               | 0m00.19s || +0m00.01s
0m00.20s | categories/Adjoint/Composition/LawsTactic                          | 0m00.17s || +0m00.03s
0m00.19s | categories/Utf8                                                    | 0m00.18s || +0m00.01s
0m00.19s | categories/Category/Objects                                        | 0m00.19s || +0m00.00s
0m00.18s | Conjugation                                                        | 0m00.14s || +0m00.03s
0m00.18s | categories/FunctorCategory/Utf8                                    | 0m00.20s || -0m00.02s
0m00.18s | categories/Functor/Utf8                                            | 0m00.17s || +0m00.00s
0m00.18s | categories/Notations                                               | 0m00.17s || +0m00.00s
0m00.18s | types/Bool                                                         | 0m00.18s || +0m00.00s
0m00.18s | categories/NaturalTransformation/Utf8                              | 0m00.15s || +0m00.03s
0m00.18s | categories/FunctorCategory/Core                                    | 0m00.18s || +0m00.00s
0m00.17s | categories/Limits/Limits                                           | 0m00.16s || +0m00.01s
0m00.17s | categories/Functor/Prod/Prod                                       | 0m00.16s || +0m00.01s
0m00.17s | categories/SetCategory/Functors/Functors                           | 0m00.20s || -0m00.03s
0m00.17s | categories/Comma/Comma                                             | 0m00.17s || +0m00.00s
0m00.17s | categories/CategoryOfGroupoids                                     | 0m00.20s || -0m00.03s
0m00.17s | categories/NatCategory                                             | 0m00.21s || -0m00.03s
0m00.17s | categories/PseudonaturalTransformation/PseudonaturalTransformation | 0m00.16s || +0m00.01s
0m00.17s | categories/HomotopyPreCategory                                     | 0m00.20s || -0m00.03s
0m00.17s | categories/Pseudofunctor/Pseudofunctor                             | 0m00.16s || +0m00.01s
0m00.17s | categories/SetCategory/SetCategory                                 | 0m00.18s || -0m00.00s
0m00.16s | categories/Profunctor/Representable                                | 0m00.18s || -0m00.01s
0m00.16s | categories/KanExtensions/KanExtensions                             | 0m00.13s || +0m00.03s
0m00.16s | categories/Structure/Structure                                     | 0m00.12s || +0m00.04s
0m00.16s | categories/Category/Univalent                                      | 0m00.12s || +0m00.04s
0m00.16s | Modality                                                           | 0m00.16s || +0m00.00s
0m00.16s | categories/GroupoidCategory/Core                                   | 0m00.16s || +0m00.00s
0m00.16s | categories/NaturalTransformation/Core                              | 0m00.15s || +0m00.01s
0m00.15s | categories/Category/Utf8                                           | 0m00.12s || +0m00.03s
0m00.15s | categories/NaturalTransformation/NaturalTransformation             | 0m00.21s || -0m00.06s
0m00.15s | categories/Profunctor/Core                                         | 0m00.11s || +0m00.03s
0m00.15s | categories/InitialTerminalCategory/Core                            | 0m00.15s || +0m00.00s
0m00.15s | categories/FunctorCategory/FunctorCategory                         | 0m00.16s || -0m00.01s
0m00.15s | hit/unique_choice                                                  | 0m00.12s || +0m00.03s
0m00.15s | categories/InitialTerminalCategory/Notations                       | 0m00.17s || -0m00.02s
0m00.15s | categories/Category/Core                                           | 0m00.12s || +0m00.03s
0m00.15s | categories/Functor/Pointwise/Pointwise                             | 0m00.11s || +0m00.03s
0m00.14s | categories/Cat/Cat                                                 | 0m00.15s || -0m00.00s
0m00.14s | categories/ExponentialLaws/Law4/Law4                               | 0m00.12s || +0m00.02s
0m00.14s | categories/Profunctor/Profunctor                                   | 0m00.16s || -0m00.01s
0m00.14s | categories/Adjoint/Composition/Composition                         | 0m00.12s || +0m00.02s
0m00.14s | categories/NaturalTransformation/Notations                         | 0m00.15s || -0m00.00s
0m00.14s | hit/TwoSphere                                                      | 0m00.13s || +0m00.01s
0m00.14s | types/Unit                                                         | 0m00.11s || +0m00.03s
0m00.14s | categories/ExponentialLaws/Law2/Law2                               | 0m00.11s || +0m00.03s
0m00.14s | Overture                                                           | 0m00.17s || -0m00.03s
0m00.14s | categories/Category/Pi                                             | 0m00.14s || +0m00.00s
0m00.14s | categories/Profunctor/Utf8                                         | 0m00.13s || +0m00.01s
0m00.13s | hit/Interval                                                       | 0m00.15s || -0m00.01s
0m00.13s | hit/iso                                                            | 0m00.17s || -0m00.04s
0m00.13s | Misc                                                               | 0m00.12s || +0m00.01s
0m00.13s | categories/Adjoint/Notations                                       | 0m00.12s || +0m00.01s
0m00.13s | categories/Category/Category                                       | 0m00.14s || -0m00.01s
0m00.13s | categories/Category/Notations                                      | 0m00.14s || -0m00.01s
0m00.13s | categories/Profunctor/Identity                                     | 0m00.12s || +0m00.01s
0m00.13s | categories/ExponentialLaws/Law1/Law1                               | 0m00.18s || -0m00.04s
0m00.12s | categories/Comma/Utf8                                              | 0m00.16s || -0m00.04s
0m00.12s | categories/Adjoint/Core                                            | 0m00.13s || -0m00.01s
0m00.12s | categories/Adjoint/Utf8                                            | 0m00.13s || -0m00.01s
0m00.12s | categories/SetCategory/Core                                        | 0m00.13s || -0m00.01s
0m00.12s | categories/Profunctor/Notations                                    | 0m00.17s || -0m00.05s
0m00.12s | categories/Category/Subcategory/Subcategory                        | 0m00.10s || +0m00.01s
0m00.12s | FunextVarieties                                                    | 0m00.15s || -0m00.03s
0m00.11s | categories/DiscreteCategory                                        | 0m00.12s || -0m00.00s
0m00.11s | categories/Functor/Core                                            | 0m00.10s || +0m00.00s
0m00.11s | categories/Comma/Notations                                         | 0m00.16s || -0m00.05s
0m00.11s | categories/Functor/Composition/Core                                | 0m00.10s || +0m00.00s
0m00.11s | categories/Category/Subcategory/Full                               | 0m00.14s || -0m00.03s
0m00.11s | hit/Truncations                                                    | 0m00.11s || +0m00.00s
0m00.11s | types/Empty                                                        | 0m00.09s || +0m00.02s
0m00.11s | categories/CategoryOfSections/CategoryOfSections                   | 0m00.15s || -0m00.03s
0m00.10s | categories/GroupoidCategory/GroupoidCategory                       | 0m00.09s || +0m00.01s
0m00.10s | Contractible                                                       | 0m00.10s || +0m00.00s
0m00.10s | categories/Category/Dual                                           | 0m00.15s || -0m00.04s
0m00.10s | categories/Category/Subcategory/Wide                               | 0m00.10s || +0m00.00s
0m00.10s | categories/Structure/Notations                                     | 0m00.11s || -0m00.00s
0m00.10s | categories/FunctorCategory/Notations                               | 0m00.10s || +0m00.00s
0m00.09s | categories/Functor/Identity                                        | 0m00.06s || +0m00.03s
0m00.09s | coq/Init/Peano                                                     | 0m00.12s || -0m00.03s
0m00.09s | types/UniverseLevel                                                | 0m00.05s || +0m00.03s
0m00.09s | categories/Structure/Utf8                                          | 0m00.10s || -0m00.01s
0m00.08s | Pointed                                                            | 0m00.06s || +0m00.02s
0m00.08s | categories/NaturalTransformation/Composition/Composition           | 0m00.08s || +0m00.00s
0m00.08s | categories/Category/Sigma/Sigma                                    | 0m00.11s || -0m00.03s
0m00.08s | coq/Program/Tactics                                                | 0m00.05s || +0m00.03s
0m00.07s | categories/Category/Strict                                         | 0m00.08s || -0m00.00s
0m00.07s | UnivalenceAxiom                                                    | 0m00.10s || -0m00.03s
0m00.07s | FunextAxiom                                                        | 0m00.06s || +0m00.01s
0m00.06s | coq/Init/Datatypes                                                 | 0m00.06s || +0m00.00s
0m00.05s | coq/Init/Notations                                                 | 0m00.04s || +0m00.01s
0m00.05s | coq/Init/Specif                                                    | 0m00.06s || -0m00.00s
0m00.05s | coq/Init/Logic_Type                                                | 0m00.07s || -0m00.02s
0m00.04s | coq/Init/Prelude                                                   | 0m00.04s || +0m00.00s
0m00.03s | coq/Init/Logic                                                     | 0m00.02s || +0m00.00s
0m00.03s | coq/Bool/Bool                                                      | 0m00.02s || +0m00.00s
0m00.03s | coq/Init/Tactics                                                   | 0m00.03s || +0m00.00s
Use [!], not [one], at Bas Spitters' suggestion
Merge pull request #407 from JasonGross/mattam82-pushout

Port Matthieu's code formalizing pushouts
I am merging, I think we have discussed enough.
WIP
WIP
WIP