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

Let $(\mathcal{C},\otimes,I_\mathcal{C})$ and $(\mathcal{D},\otimes,I_\mathcal{D})$ be left-closed monoidal categories with internal-hom denoted by $[X,Y]$ and $F:\mathcal{C}\to\mathcal{D}$ be a monoidal functor.

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

As a motivating example, observe the case where $\mathcal{C}$ is a Cartesian closed category of topological spaces and $F:\mathcal{C}\to\mathbf{Set}$ is the forgetful functor.

Answer

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