$U$-split pairs and ordinary or strict creation of their coequalizers

Let $U:\mathcal{D}\to \mathcal{C}$ be a functor. A parallel pair $f,g:d\rightrightarrows d^{\mathrm{\prime }}$ in $\mathcal{D}$ is $U$-split when its image under $U$ extends to a split coequalizer diagram in $\mathcal{C}$ (Split coequalizer diagrams).

The functor $U$ creates coequalizers of $U$-split pairs when each chosen split coequalizer of $Uf$ and $Ug$ is isomorphic, as a coequalizer diagram, to the image under $U$ of a coequalizer of $f$ and $g$, and every lifted fork whose image is a coequalizer is itself a coequalizer. This is ordinary isomorphism-invariant creation in the sense of Preservation, reflection, and creation of limits and colimits; continuous and cocontinuous functors.

The functor $U$ strictly creates coequalizers of $U$-split pairs when every supplied splitting has a unique lift on the same apex and legs, and the lifted fork is a coequalizer. Strict creation therefore specifies the lifted object and structure on the nose, rather than only up to isomorphism.