### Monoidal ends

Let $(\mathcal{C},\otimes)$ be a monoidal category and let $(\mathcal{D},\times)$ be a *cartesian* monoidal category. Let $(X,\eta,\mu) : (\mathcal{C}^{\mathrm{op}},\otimes) \times (\mathcal{C},\otimes) \to (\mathcal{D},\times)$ be a lax monoidal functor. A wedge to $(X,\eta,\mu)$ is a wedge to the underlying functor $X : \mathcal{C}^{\mathrm{op}} \times \mathcal{C} \to \mathcal{D}$, thus consisting of an object $T \in \mathcal{D}$ and a family of morphisms $(w_A : T \to X(A,A))_{A \in \mathcal{C}}$, such that the following properties hold:

$w_1 : T \to X(1,1)$ is equal to
$$T \xrightarrow{\exists!} 1 \xrightarrow{\eta} X(1,1).$$

$w_{A \otimes B} : T \to X(A \otimes B, A \otimes B)$ is equal to
$$T \xrightarrow{~(w_A,w_B)~} X(A,A) \times X(B,B) \xrightarrow{~~\mu~~} X(A \otimes B, A \otimes B).$$

A universal wedge, i.e. end, is defined as usual. It is easy to see that if $\mathcal{C}$ is small and $\mathcal{D}$ is complete, then any lax monoidal functor has an end. We can denote it by $\int (X,\eta,\mu)$.

Since it is a common practice (*sigh*) to ignore forgetful functors and just write $X$ both for the functor and the monoidal functor, some people will prefer to call this a "monoidal wedge" and a "monoidal end", the latter then being denoted by something like $\int^{\otimes} X$. I do not know if this concept has appeared elsewhere, I just made it up to answer the question below.

## Monoidal natural transformations

Now let $(\mathcal{C},\otimes)$, $(\mathcal{C}',\otimes)$ be two monoidal categories and $(F,\eta_F,\mu_F),(G,\eta_G,\mu_G) : (\mathcal{C},\otimes) \to (\mathcal{C}',\otimes)$ be two strong monoidal functors. Consider the functor
$X : \mathcal{C}^{\mathrm{op}} \times \mathcal{C} \to \mathbf{Set}$ defined on objects by
$$X(A,B) := \mathrm{Hom}(F(A),G(B)).$$
We equip it with the following lax monoidal structure:
$$\eta_X : 1 \to X(1,1)$$
corresponds to the isomorphism $\eta_G \circ \eta_F^{-1} : F(1) \to 1 \to G(1)$, and
$$\mu_X : X(A,A') \times X(B,B') \to X(A \otimes B,A' \otimes B')$$
maps a pair of morphisms $f : F(A) \to G(A')$, $g : F(B) \to G(B')$ to the morphism
$$F(A \otimes B) \xrightarrow{\mu_F^{-1}} F(A) \otimes F(B) \xrightarrow{f \otimes g} G(A') \otimes G(B') \xrightarrow{\mu_G} G(A' \otimes B').$$
One needs to check the coherence conditions in the definition of a lax monoidal functor, I will not do this here.

It is straight forward to check that $\int (X,\eta,\mu)$ is the set of morphisms $(F,\eta_F,\mu_F) \to (G,\eta_G,\mu_G)$ (aka monoidal natural transformations).

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