# Epimorphism and image

Let $\phi:A\rightarrow B$ be a morphism in a category, and $\phi':I\hookrightarrow B$ its image. Intuitively, $\phi$ should be an epimorphism if $\phi'$ is an epimorphism. But I have difficulty proving it using definitions of image and epimorphism (from Wikipedia). Is it true at all?

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Here is a counterexample. Let us have a category with five objects $A, B, C, D, E$, and non-identity arrows $A \rightrightarrows B$, $B \to C$, $C \to D$, $D \rightrightarrows E$, such that the composites $C \to D \rightrightarrows E$ are distinct while the pairs of composites $B \to C \to D \rightrightarrows E$ and $A \rightrightarrows B \to C$ are not distinct. By construction, $B \to C$ is not a monomorphism, $C \to D$ is a monomorphism and is the image of $B \to D$, while $C \to D$ is an epimorphism but $B \to D$ is not.