Let (C,⊗,IC) and (D,⊗,ID) be left-closed monoidal categories with internal-hom denoted by [X,Y] and F:C→D be a monoidal functor.

It is straightforward to see that there are morphisms ϕ:F([X,Y])→[F(X),F(Y)] in D, which are natural in X and Y. What are sufficient conditions on F (or possibly the categories in question) to ensure that ϕ is a monomorphism for all X,Y in C?

As a motivating example, observe the case where C is a Cartesian closed category of topological spaces and F:C→Set is the forgetful functor.

**Answer**

**Attribution***Source : Link , Question Author : Jeremy Brazas , Answer Author : Community*