@jonmsterling some ideas but not fleshed out to usable slogans:
draw a function via its cograph, that is the formalization of “potato diagrams”, where we have a set A on the left, a set B on the right and an edge from any element a to where it is sent. From this we can take the mirror image diagram (reverse all edges) and ask if this is a function - the inverse function
draw the graph of a function and consider the reflection in the diagonal (lots of people will be used to this idea as log is the “mirror image” relation of exp or sqrt and square functions etc.) - this relation may also be functional, and is the inverse
more speculative: the ur-family of isomorphisms arise from the first isomorphism theorem of sets, a set quotiented by the kernel pair equiv relation is iso to the image is iso to the equalizer of the cokernel pair. I feel like there must be some way to make this idea palatable to get a deeper perspective e.g. this describes the iso between Z/nZ and [n], but this is notoriously hard to get across in an intro course