Morphisms

General maps

one-to-one	injective
onto	surjective
any map	bijective

Linear maps

any map	homomorphism
one-to-one	monomorphism
onto	epimorphism
one-to-one and onto	isomorphism

Linear self-maps

any map	endomorphism
one-to-one and onto	automorphism

Examples of isomorphisms and maps that identifies the isomorphism:

• $M_{1,3}(\Bbb F) \cong M_{3,1}(\Bbb F)\\ (x_1,x_2,x_3)\mapsto \begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}$


• $M_{2,2}(\Bbb F) \cong \Bbb F^4\\ \begin{pmatrix}a&b\\ c&d \end{pmatrix}\mapsto (a,b,c,d)$


• $M_{2,3}(\Bbb F) \cong M_{3,2}(\Bbb F)\\ A\mapsto A^T$


• The plane $z=0$ in $\Bbb R^3$ is isomorphic to $\Bbb R^2$.
  $(x,y,0)\mapsto (x,y)$


• $P_n\cong \Bbb R^{n+1}\\ a_0+a_1x+\cdots+a_nx^n\mapsto (a_0,a_1,\cdots,a_n)$


• $P$ is isomorphic to $\Bbb R_0^\infty$.
  $a_0+a_1x+\cdots+a_nx^n\mapsto (a_0,a_1,\cdots,a_n,0,0,\cdots)$


• $M_{m,n}(\Bbb F) \cong L(\Bbb F^n, \Bbb F^m)$
  $A\mapsto L_A$, where $L_A(x)=Ax$.


• Any vector space $V$ of dimension $n$ is isomorphic to $\Bbb F^n$.
  $v\mapsto [v]_\alpha$, where $\alpha$ is a basis for $V$.

Category theory
A morphism $f:A\to B$ in a category $C$ is an isomorphism if there exists a morphism $g:B \to A$ such that both ways to compose $f$ and $g$ give the identity morphisms on the respective objects, namely $$gf=1_A, \quad fg=1_B.$$
A morphism $f:A\to B$ in a category $C$ is an monomorphism if for every object $C$ and every pair of morphisms $g,h:C\to A$, the condition $fg=fh$ implies $g=h$. Namely, the following diagram commutes:

When we say a diagram commutes, it means all ways to compose morphisms result in an identical morphisms. Therefore, whenever $f$ is an monomorphism and this diagram commutes, we can conclude $g=h$. The key idea is that monomorphisms allow us “cancel” $f$ from the left side of a composition.

The corresponding property for cancelling on the right is defined similarly.

A morphism $f:A\to B$ in a category $C$ is an epimorphism if for every object $C$ and every pair of morphisms $g,h:B\to C$, the condition $gf=hf$ implies $g=h$.

Whenever $f$ is an epimorphism and this diagram commutes, we can conclude $g=h$.

Reference:
Examples of isomorphisms
Properties of morphisms
An awesome video on morphisms