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I am interested in a certain category $\mathcal D$. $\mathcal D$ is complete and cocomplete. There is a symmetric monoidal structure $(\mathcal D,\otimes,I)$ that was introduced a while ago, but it is still not well understood. The unit object $I$ is initial, hence $\mathcal D$ is a semicocartesian monoidal: every $A\otimes B$ is equipped with a pair of coprojections $j_A\colon A \to A\otimes B$ and $j_B\colon B\to A\otimes B$.

What I am trying to do now is to characterize/construct the tensor product $\otimes$ in various ways, using as much abstract nonsense as possible.

I have come up with the following construction:

There is an essentially small dense subcategory of $\mathcal D$ which I will denote $\mathcal C$. Unlike the category $\mathcal D$, the category $\mathcal C$ is well understood and nice. It is finitely complete and cocomplete, co-exponentially closed. The embedding of $\mathcal C$ into $\mathcal D$ is full and faithful, preserves finite limits and the initial objects of $\mathcal C$ and $\mathcal D$ coincide.

There is an obviously defined functor $R:\mathcal D\to [\mathcal C^{op},\mathbf{Set}]$ that takes an object $A$ of $\mathcal D$ to the presheaf $\mathcal D(\_,A):\mathcal C^{op}\to{\mathbf{Set}}$. Since $\mathcal D$ is cocomplete, $R$ is a right adjoint for general reasons (see page 41 result of this search). The corresponding left adjoint maps a presheaf $P$ on $\mathcal C$ to the colimit of the "projection" functor $\int P\to\mathcal C\hookrightarrow\mathcal D$, where $\int P$ is the category of elements of the presheaf $P$. In this case it can be proved that the adjunction is a reflection, hence $\mathcal C$ is dense in $\mathcal D$.

Let $A,B$ be two objects of $\mathcal D$. Consider the functor $$ F_{A,B}:\int R(A)\times\int R(B)\to\mathcal D $$ given by the rule $F_{A,B}((X_A,g_A),(X_B,g_B))=X_A*X_B$, where $*$ denotes the coproduct in $\mathcal C$ (this is not the same thing as the coproduct in $\mathcal D$).

I have proved the following: $$ \varinjlim F_{A,B}\simeq A\otimes B $$

So, in this particular case, the construction outlined above is a functor part of a symmetric monoidal structure on $\mathcal D$. However, to prove this fact I need to use internal structure of the objects of $\mathcal D$.

Question: Is there an abstract way how to finish the proof of the fact that this construction gives us a symmetric monoidal structure on $\mathcal D$?

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    $\begingroup$ Don’t have time to write a full answer right now, but a keyword here is Day convolution — it looks like you’re taking the Day convolution of the coproduct of C, which gives a symmetric monoidal structure on Psh(C), and then reflecting this into D. $\endgroup$ – Peter LeFanu Lumsdaine Mar 30 '17 at 13:28
  • $\begingroup$ @PeterLeFanuLumsdaine Hmm. If I did not flip the arrows (again!), the colimit in the Day convolution you mentioned contains something like "arrows from a fixed object of $\mathcal C$ into all possible coproducts of objects of $\mathcal C$". Arrows into coproducts are ... strange. But $\mathcal C$ is co-exponentially closed, maybe that helps. $\endgroup$ – Gejza Jenča Mar 30 '17 at 15:12

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