@Anindya - thank you, indeed, a nice observation. So restating it - if function \$$f\$$ has a left inverse \$$g\$$ and a right inverse \$$h\$$, then these inverses are equal (isomorphic), and \$$f\$$, in turn, has the unique inverse.

This translates to the following:

- \$$f\$$ having a left inverse is another way to state that it is injective

- \$$f\$$ having a right inverse is another way to state that it is surjective

- having both thus translates to that \$$f\$$ is bijective, and thus has the unique inverse \$$f^{-1}\$$.

**EDIT:**

But actually these 2 properties is much more general and abstract way to think about morphisms not just as about functions between sets, and this innocuously looking lecture brought a lot of surprising things to our attention.

I use the following mnemonic: having a left inverse means that there is no loss of information when morphing an object \$$x\$$ to an object \$$y\$$, so we can somehow reverse this process and get back to \$$x\$$.

And having a right inverse allows us to reconstruct \$$y\$$, previously morphed into \$$x\$$ by \$$g: y \to x\$$.

It is interesting that using these we can relax the requirement \$$g \circ f = 1_x\$$ by defining some equivalence classes on \$$x\$$ and allowing the identity morphism to hold up to isomorphism - i.e. we can partially restore \$$x\$$ and \$$y\$$ both ways. Actually these are the cases which I'm interested in the most, looking forward what comes next in the course.