You can see the kind of triangle I'm talking about here:

* nLab, [Under category](https://ncatlab.org/nlab/show/under+category).

We say a function \$$f : A \to P\$$ is an object in the category of **sets under \$$A\$$**. Given two sets under \$$A\$$, say \$$f : A \to P\$$ and \$$g: A \to Q\$$, we say a morphism from one to the other is a function \$$h: P \to Q\$$ such that \$$h \circ f = g\$$. If \$$h\$$ is an isomorphism, we say \$$f : A \to P\$$ and \$$g: A \to Q\$$ are isomorphic in the category of sets under \$$A\$$.