 Solution

A congruence on a category is an equivalence relation on every hom-set such that $f\sim f^{\prime }$ and $g\sim g^{\prime }$ imply $gf\sim g^{\prime }{f}^{\prime }$ whenever the composites exist. The quotient has the same objects and equivalence classes $[f]$ as morphisms.

For the proposed $\Phi$-maps, reflexivity uses $W=U$, symmetry is immediate, and transitivity is obtained as follows. If representatives over $U$ and $V$ agree after restriction to $W$, while those over $V$ and $T$ agree after restriction to $W^{\prime }$, then their first and third representatives agree over $W\times W^{\prime }$. This object belongs to $\Phi$ and maps below $U\times T$, proving transitivity.

The identity of $A$ is represented by the projection $A\times 1\to A$. If $f:A\times U\to B$ and $g:B\times V\to C$, define their composite over $U\times V$ byPassing to a smaller member of $\Phi$ shows that this is independent of representatives. Associativity follows from associativity of products and composition, and the projection representatives satisfy the identity laws. This constructs the category of partial maps localized at subterminal objects ${\mathcal{C}}_{\Phi }$.

The terminal object remains $1$. Products are the products of $\mathcal{C}$: representatives $f:C\times U\to A$ and $g:C\times V\to B$ pair after restriction to $U\times V$,The product universal property follows after restricting competing representatives to a common member of $\Phi$. The functor $P_{\Phi }$ sends the original projections and pairings to these, so it preserves finite products.

If $\mathcal{C}$ is Cartesian closed, use the same exponential object $B^A$. A representativemay be rearranged as $(C\times U)\times A\to B$ and curried in $\mathcal{C}$ to $C\times U\to B^A$. Currying respects restriction and gives a natural bijectionThus ${\mathcal{C}}_{\Phi }$ is cartesian closed and $P_{\Phi }$ preserves exponentials.

In general this is not a quotient by a congruence. A congruence can identify existing parallel morphisms but cannot create a morphism between two objects. Take $\mathcal{C}=\mathbf{Set}$ and $\Phi =\{0,1\}$, the filter containing the empty subobject of the terminal set. Then every $A\times 0\to B$ represents a $\Phi$-map, so in particular ${\mathcal{C}}_{\Phi }(A,\varnothing )$ is nonempty for nonempty $A$, whereas $\mathbf{Set}(A,\varnothing )$ is empty. Hence no quotient of $\mathbf{Set}$ by a congruence is isomorphic to this ${\mathcal{C}}_{\Phi }$ by an identity-on-objects functor.