Canonical morphism $F(X,I)\otimes Y\to F(X,Y)$ in a closed symmetric monoidal category

Question

Let $(\mathcal{C},\otimes,I,F)$ be a closed symmetric monoidal category, where $F$ is the 'internal Hom' functor, $I$ is the unit object and $\otimes$ is the monoidal product.

I am reading Definition 1.1.2 of Axiomatic Stable Homotopy Theory, which refers to the 'natural map' $F(X,I)\otimes Y\to F(X,Y)$ (it is in topological notation in the text). How do I construct such a morphism from the properties of $\mathcal{C}$?

Answer 1

I will denote the internal hom by $[X,Y]$. It is defined by an adjunction $$\alpha_{X,Y,T} : \hom(T,[X,Y]) \cong \hom(X \otimes T,Y),$$ and in particular it comes equipped with a counit morphism $$\varepsilon_{X,Y} : X \otimes [X,Y] \to Y,$$ namely $\varepsilon_{X,Y} := \alpha_{X,Y,[X,Y]}(\mathrm{id}_{[X,Y]})$.

For any three objects $X,Y,Z$ we then have the morphism $$\varepsilon_{X,Z} \otimes \mathrm{id}_Y : X \otimes [X,Z] \otimes Y \to Z \otimes Y$$ and therefore $$\omega_{X,Y,Z} := \alpha_{X,Z \otimes Y,[X,Z] \otimes Y}^{-1}(\varepsilon_{X,Z} \otimes \mathrm{id}_Y) : [X,Z] \otimes Y \to [X,Z \otimes Y]$$ For $Z=1$ we get $\omega_{X,Y,1} : [X,1] \otimes Y \to [X,Y]$.

Notice that $\omega_{X,Y,Z}$ can be described in element notation by $\omega_{X,Y,Z}(f \otimes y) = \bigl(x \mapsto f(x) \otimes y\bigr)$.