I recall reading some time ago about some pattern/structure in category theory. Now I need to study some related properties and can't recall the proper name of it.
Let me describe it (pardon me if I am not 100% precise with my notation).
Let A be monoid over set a, with identity element a_id and binary assoc. operation:
(a op_a a) -> a
Likewise B is a monoid over set b, with identity element b_id and binary assoc. operation:
(b op_b b) -> a
What is the name of structure S, that consists of:
AB- mapping
m1:a -> b - mapping
m2:b -> a
... such that:
(a1 op_a a2) = m2(m1(a1) op_b m2(a1))
... for all a1, a2 in a?
Example of this in math is:
exp(log(a1) + log(a2)) = a1 * a2
... with A and B being monoids over rational numbers with multiplication and addition operations, m1 being log and m2 being exp.
Another S-like structure example is when you define mappings for integers <-> strings (where strings are limited to be repetition of some symbol n times) and sum and concat forming monoids.
So, what is the proper name for this structure in category theory (or other branches of math)?
I was looking for group isomorphism. Kudos to commenters for pointing me in right direction.