It is well known (e.g., Reference for "lax monoidal functors" = "monoids under Day convolution" ) that if $ \mathcal C$ is a monoidal $ \mathcal V$ -enriched category, then a monoid in $ [\mathcal C, \mathcal V]$ is the same thing as a lax monoidal functor $ \mathcal C \to \mathcal V$ , where $ [\mathcal C,\mathcal V]$ carries the monoidal structure given by the Day convolution.

Is the following also true?

If $ \mathcal C,\mathcal D$ are monoidal $ \mathcal V$ -enriched categories, where $ \mathcal V$ is cocomplete, then a lax monoidal functor $ \mathcal C \to [\mathcal D,\mathcal V]$ is the same thing as a lax monoidal functor $ \mathcal C \otimes \mathcal D \to \mathcal V$ .

(In particular, if $ \mathcal C$ is the unit category, then we get the original formulation.)

It certainly seems to be the case: if $ \mathcal F \colon \mathcal C \otimes \mathcal D \to \mathcal V$ is a lax monoidal functor, then we have natural coherences \begin{gather} \mathcal F(c,\_)\otimes_{\text{Day}}\mathcal F(c’,\_) \to \mathcal F(c\otimes c’,\_) \ I_{\text{Day}} \to F(I, \_)\,, \end{gather} where the multiplicative coherence is given by the composite \begin{align} & \int^{d,d’\colon\mathcal D}(\mathcal F(c,d)\otimes\mathcal F(c’,d’))\otimes \mathcal D(d\otimes d’,x)\ \to & \int^{d,d’\colon\mathcal D}\mathcal F(c\otimes c’,d\otimes d’)\otimes\mathcal D(d\otimes d’,x)\ \to & \mathcal F(c\otimes c’,x)\,. \end{align} and the unital coherence $ \mathcal D(I_{\mathcal D},x)\to\mathcal F(I_{\mathcal C},x)$ comes (via the enriched Yoneda lemma) from the monoidal unit $ I_{\mathcal V}\to\mathcal F(I_{\mathcal C},I_{\mathcal D})$ for $ \mathcal F$ .

I have not checked whether these satisfy the coherence conditions for a monoidal functor, but I would be surprised if they did not.

In the other direction, if $ \mathcal F\colon \mathcal C \to [\mathcal D,\mathcal V]$ is lax monoidal, then we have coherences given by \begin{align} &\mathcal G(c)(d) \otimes \mathcal G(c’)(d’)\ \cong & \int^{e,e’\colon\mathcal D} (\mathcal G(c)(e) \otimes \mathcal G(c’)(e’)) \otimes (\mathcal D(e,d) \otimes \mathcal D(e’,d’)) \ \to & \int^{e,e’\colon\mathcal D} (\mathcal G(c)(e) \otimes \mathcal G(c’)(e’)) \otimes \mathcal D(e\otimes e’,d\otimes d’)\ = & (\mathcal G(c) \otimes_{\text{Day}}\mathcal G(c’))(d \otimes d’)\ \to &\mathcal G(c\otimes c’)(d\otimes d’) \end{align} and monoidal unit $ I_{\mathcal V}\to \mathcal F(I_{\mathcal C},I_{\mathcal D})$ by the enriched Yoneda lemma as before.

I haven’t checked whether these coherences actually satisfy the appropriate diagrams, nor whether these two maps are indeed inverses.

Is this fact true? And if so, has it been proved in the literature somewhere?