@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.

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.