Do the forgetful functors $G_H:\bf Monoid \to \bf Semigroup^1$ and $G_O:\bf Semigroup^1 \to \bf Semigroup$ have left and/or right adjoints? Here $\bf Semigroup$ is the category of semigroups with semigroup homomorphisms, $\bf Semigroup^1$ is the category of monoids (semigroups with identity) with semigroup homomorphisms, and $\bf Monoid$ is the category of monoids with monoid homomorphisms (the image of the identity is the identity).

An interesting functor $F_H:\bf Semigroup^1 \to \bf Monoid$ is given by the functor which leaves the monoid objects unchanged, and modifies the homomorphisms by replacing the image of the identity by the identity of the target monoid. Is this a left or right adjoint functor to $G_H$?

A nice functor $F_O:\bf Semigroup \to \bf Semigroup^1$ is the one which leaves monoids unchanged, and adjoins an identity element to semigroups missing an identity element. The homomorphisms are extended by mapping the newly adoint identity element to the identity element of the target monoid. Is this a left or a right adjoint functor to $G_O$, or both?

A well known functor $F:\bf Semigroup \to \bf Monoid$ is the one which adjoins to each semigroup an identity (independent of whether it already has one), and extends the homomorphisms by mapping the newly adoint identity element to the newly adjoint identity element of the target monoid. I have seen the assumption (statement) that this is the left adjoint of the forgetful functor. Is this correct?


Question: Installing Windows 10 in VirtualBox and in MBM
Answer: You should use (for example) application as: Acronis True Image or Norton Ghost or another image creating 

Run this program in VBox session (in Windows 10 guest) from start CD/image.

Make image this guest (for example) to network disk.

Next - restore these images to clean disk.

And run computer from this disk.

It should be run :-). But maybe is not run :-(.