Suppose I have a finite graph $G$, and I then take the free category $\mathcal{C}(G)$ over such a finite graph. Now, I would like to "force" some objects to be limits. Is there a way to do that simply, or do I need to be really careful with the relations I take ?

Ex: Let $G$ be the graph with objects $1$, $A$, $B$, $A\times B$ and let us add "projection edges" $p_1: A\times B \rightarrow A$, $p_2: A\times B \rightarrow B$. We also add one edge from $A$, $A\times B$ and $B$ to $1$. Taking the free category over $G$ and quotienting by the congruence identifying all arrows with codomain $1$ will make $1$ a terminal object. Yet, $A \times B$ won't be a product here because the diagonal map $\Delta : A \rightarrow A \times B$ doesn't exist (similarly for $B$). So I should put it by hand on $G$ if I wanted to have a product at the end of this construction. On the contrary, adding "too much" arrows at the beginning of this construction and forcing that for any given arrows that factor through $A \times B$ there exists a unique map with codomain $A \times B$ doesn't help me to identify the second factor.

Is there a nice way to force some objects in a presentation to be limits?

What I want is to be able to produce some archetype of syntactic category, say, the category that contains only products of $A$ and $B$ and their universal arrows between them. Then, I would like to add some non trivial map, say, $m : A\times A \rightarrow A$, such that

1) it does not break any products

2) all composite involving $m$ coming from universal arrows exist.

Some idea: Let $J : \mathcal{D} \rightarrow \mathcal{C}$ be a diagram. We say that an object of $\mathcal{C}$ is $J$-elementary if it is not a limit object of $J$. An arrow of is said to be $J$-elementary if it is not a limit of $J$ (inside the limit cone diagram) neither an universal arrow for $J$.

A category is said to be $J$-generated if :

1) any $J$-elementary object admits a $J$-limit,

2) the only $J$-elementary arrows of non $J$-elementary object are identities.

A category is said to be $J$-generated $J$-closed if it is $J$-generated and if any objects admits a $J$-limit.

Can this idea works, and if yes, is there a universal property for it? Still, if it does, I think that there's no way to add arrows without breaking the product structure when $J$ is say, the diagram of two objects.

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    $\begingroup$ It seems to me that you are trying to reinvent sketches and normal sketches. $\endgroup$ – Zhen Lin Apr 2 '15 at 23:00
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    $\begingroup$ What Zhen Lin says sounds about right to me. A good place to read about this is Adamek and Rosicky's book, Locally Presentable and Accessible Categories. $\endgroup$ – Todd Trimble Apr 2 '15 at 23:30

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