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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.

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    $\begingroup$ I don't think that there is such a condition. Consider for example the special case of module categories and a cocontinuous symmetric monoidal functor. Here the question is: For which homomorphisms of commutative rings $R \to S$ is the natural map $\hom_R(M,N) \otimes_R S \to \hom_S(M \otimes_R S, N \otimes_R S)$ injective for all $R$-modules $M,N$? There are some trivial conditions when this map is actually an isomorphism, but I don't know of any conditions where it is just a monomorphism. $\endgroup$
    – HeinrichD
    Apr 22, 2017 at 14:56
  • $\begingroup$ I don't doubt there are many situations where $\phi$ is not a monomorphism but there are definitely some. For example, the forgetful functor $\mathbf{CG}\to \mathbf{Set}$ from compactly generated spaces to sets has this property. Faithfulness may be a start. $\endgroup$
    – Jeremy Brazas
    Apr 22, 2017 at 15:08

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