Let $\mathcal{C}$ be a category. A monomorphism $f:A\to B$ in $\mathcal{C}$ is said to be a strong monomorphism if, whenever we are given the following commutative diagram $$\xymatrix@+=3pc{ {C}\ar[r]^{g}\ar[d]_{x}&{D}\ar[d]^{y}\\ {A}\ar[r]_{f}&{B} } $$ with $g$ an epimorphism, then there is a morphism $h: D\to A$ such that the following is another commutative diagram: $$\xymatrix@+=3pc{ {C}\ar[r]^{g}\ar[d]_{x}&{D}\ar[d]^{y} \ar@{.>}[dl]|{h} \\ {A}\ar[r]_{f}&{B} } $$

Note that the ``diagonal'' morphism $h$ is necessarily unique. In other words, a monomorphism is strong iff every epimorphism is orthogonal to it.

Dually, a strong epimorphism is an epimorphism which is orthogonal to every monomorphism in the category.

Remark. Every regular monomorphism is strong (see proof here), and every strong monomorphism is extremal.

Proof. Suppose $f:A\to B$ is a strong monomorphism and that $f=h\circ g$ with $g:A\to C$ epimorphic. Then we have the following commutative diagram $$\xymatrix@+=3pc{ {A}\ar[r]^{g}\ar[d]_{1_A}&{C}\ar[d]^{h}\\ {A}\ar[r]_{f}&{B} } $$ Since $f$ is strong, there is a morphism $e:C\to A$ such that the diagram below is commutative $$\xymatrix@+=3pc{ {A}\ar[r]^{g}\ar[d]_{1_A}&{C}\ar[d]^{h} \ar@{.>}[dl]|{e} \\ {A}\ar[r]_{f}&{B} } $$ This shows that $g$ is a split monomorphism, as $1_A=e\circ g$ But $g$ is epimorphic, we conclude that $g$ is an isomorphism (this fact is proved here).

Bibliography

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F. Borceux Basic Category Theory, Handbook of Categorical Algebra I, Cambridge University Press, Cambridge (1994)