= Solution
A <congruence on a category> is an equivalence relation on every hom-set such that $f\sim f'$ and $g\sim g'$ imply $gf\sim g'f'$ 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 $W'$, then their first and third representatives agree over $W\times W'$. This object belongs to $\Phi$ and maps below $U\times T$, proving transitivity.
The identity of $A$ is represented by the projection $A\times1\to A$. If $f:A\times U\to B$ and $g:B\times V\to C$, define their composite over $U\times V$ by
$$
A\times U\times V\xrightarrow{f\times1_V}B\times V\xrightarrow{g}C.
$$
Passing 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 $U\times V$,
$$
C\times U\times V\longrightarrow A\times B.
$$
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 representative
$$
f:(C\times A)\times U\to B
$$
may 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 bijection
$$
\mathcal C_\Phi(C\times A,B)\cong\mathcal C_\Phi(C,B^A).
$$
Thus $\mathcal C_\Phi$ 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 $A\times0\to B$ 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.
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