“Monomorphicness” of a natural morphism induced by a monoidal functor

Let (C,,IC) and (D,,ID) be left-closed monoidal categories with internal-hom denoted by [X,Y] and F:CD 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:CSet is the forgetful functor.

Answer

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

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