Let $\mathscr{S}$ be a [limit sketch](https://ncatlab.org/nlab/show/sketch) in a small category $\mathcal{E}$, i.e. just a collection of cones in $\mathcal{E}$. Then its category $\mathbf{Mod}(\mathscr{S})$ of models (i.e. functors $\mathcal{E} \to \mathbf{Set}$ which send the cones in $\mathscr{S}$ to limit cones) is cocomplete, in fact locally presentable. It enjoys the following universal property in the $2$-category of cocomplete categories: If $\mathcal{C}$ is any cocomplete category, then
$$\mathrm{Hom}\_c(\mathbf{Mod}(\mathscr{S}),\mathcal{C}) \simeq \mathbf{Mod}\_{\mathcal{C}}(\mathscr{S}^{\mathrm{op}}).$$
Here, $\mathrm{Hom}\_c$ denotes the category of cocontinuous functors, $\mathscr{S}^{\mathrm{op}}$ denotes the dual *colimit* sketch, and $\mathbf{Mod}\_{\mathcal{C}}$ refers to $\mathcal{C}$-valued models.


A typical example of this universal property is $\mathrm{Hom}\_c(\mathbf{Grp},\mathcal{C}) \simeq \mathbf{CoGrp}(\mathcal{C})$, that is, $\mathbf{Grp}$ is the universal example of a cocomplete category with an internal cogroup object.


The proof is a combination of two well-known results, so I assume that this universal property is also well-known. **Can someone confirm this and point me to literature which I can cite for this result?**


Maybe it follows from some of the theorems in [Kelly's book](http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html) on enriched categories, chapter 6. But I am not sure. 




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Here is a sketch of the two-step proof I was thinking of. Let $G : \mathbf{Mod}(\mathscr{S}) \to [\mathcal{E},\mathbf{Set}] $ be the inclusion and $F : [\mathcal{E},\mathbf{Set}] \to \mathbf{Mod}(\mathscr{S})$ its left adjoint. Then by a "tensor-less" variant of Prop. 2.3.6. in [Tensor functors between categories of quasi-coherent sheaves](https://arxiv.org/pdf/1202.5147.pdf) (as you see, this is not quite the reference I need!) the category $\mathrm{Hom}\_c(\mathbf{Mod}(\mathscr{S}),\mathcal{C})$ is equivalent to the category of those cocontinuous functors $Q^\* : [\mathcal{E},\mathbf{Set}] \to \mathcal{C}$ such that $Q^\* \to Q^\* \circ G \circ F$ is an isomorphism; this is true iff the right adjoint $Q\_\* : \mathcal{C} \to [\mathcal{E},\mathbf{Set}]$ factors over $\mathbf{Mod}(\mathscr{S})$. By the well-known universal property of $[\mathcal{E},\mathbf{Set}]$ as the free cocompletion of $\mathcal{E}^{\mathrm{op}}$ (for instance, Section 4.4 in Kelly's book), $Q^\*$ corresponds to a functor $P : \mathcal{E}^{\mathrm{op}} \to \mathcal{C}$ via $Q^\* = P \otimes\_{\mathcal{E}} -$, and hence $Q\_\* = \mathrm{Hom}(P(-),-)$. By the Yoneda Lemma, the requirement that $Q\_\*$ factors over $\mathbf{Mod}(\mathscr{S})$ exactly means that $P$ is a model of $ \mathscr{S}^{\mathrm{op}}$.




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I have found a reference: Theorem 2.2.4 in [this paper](https://arxiv.org/abs/1105.3104) (actually, I knew this paper, but I somehow forgot that the theorem is in there). But the authors also explicitly state that this theorem is well-known. I would be happy with a more "classical" or "original" reference.



Universal property of the cocomplete category of models of a limit sketch
Probably the earliest reference is Theorem 2.5 in



> 
> A. Pultr, **The right adjoints into the categories of relational systems**, Reports of the Midwest Category
> Seminar IV. Springer, Berlin, Heidelberg, 1970
> 
> 
> 


Pultr's "relational theories" are exactly small realized limit sketches.


Remark: I have recently generalized the result to large limit sketches ([arXiv:2106.11115](https://arxiv.org/abs/2106.11115)).


