Inserter category

In category theory, a branch of mathematics, the inserter category is a variation of the comma category where the two functors are required to have the same domain category.
Definition.
If "C" and "D" are two categories and "F" and "G" are two functors from "C" to "D", the inserter category Ins("F", "G") is the category whose objects are pairs ("X", "f") where "X" is an object of "C" and "f" is a morphism in "D" from "F"("X") to "G"("X") and whose morphisms from ("X", "f") to ("Y", "g") are morphisms "h" in "C" from "X" to "Y" such that formula_1.
Properties.
If "C" and "D" are locally presentable, "F" and "G" are functors from "C" to "D", and either "F" is cocontinuous or "G" is continuous; then the inserter category Ins("F", "G") is also locally presentable.