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Suppose whole square and the left square in the diagram below are pullbacks, then we may wonder whether the right square is a pullback. It is usually not the case.

Now we seek some addition condition on $X\to Y$ that forces the right square is a pullback too.

My question: is epic a sufficient condition? (If the category is Sets, then yes.)

Added: Let $P$ be the pullback of the right square, then there exist $B\to P$, and the square $A\to P \to Y$ // $A\to X \to Y$ is a pullback, so we have the following diagram in which the bottom and the whole squares are pullback, so is the upper square. If the category is Sets, $X\to Y$ is surjective then $A\to P $ is also surjective. Since the pullback of $B\to P$ along a surjective map is an bijection, $B\to P$ must be a bijection. This shows the right square of the original diagram is a pullback.

Suppose whole square and the left square in the diagram below are pullbacks, then we may wonder whether the right square is a pullback. It is usually not the case.

Now we seek some addition condition on $X\to Y$ that forces the right square is a pullback too.

My question: is epic a sufficient condition? (If the category is Sets, then yes.)

Added: Let $P$ be the pullback of the right square, then there exists $B\to P$, and the square $A\to P \to Y$ // $A\to X \to Y$ is a pullback, so we have the following diagram in which the bottom and the whole squares are pullback, so is the upper square. If the category is Sets, $X\to Y$ is surjective then $A\to P $ is also surjective. Since the pullback of $B\to P$ along a surjective map is an bijection, $B\to P$ must be a bijection. This shows the right square of the original diagram is a pullback. We can also see why we consider some nice condition on $X\to Y$.

Suppose whole square and the left square in the diagram below are pullbacks, then we may wonder whether the right square is a pullback. It is usually not the case.

Now we seek some addition condition on $X\to Y$ that forces the right square is a pullback too.

My question: is epic a sufficient condition? (If the category is Sets, then yes.)

Suppose whole square and the left square in the diagram below are pullbacks, then we may wonder whether the right square is a pullback. It is usually not the case.

Now we seek some addition condition on $X\to Y$ that forces the right square is a pullback too.

My question: is epic a sufficient condition? (If the category is Sets, then yes.)

Added: Let $P$ be the pullback of the right square, then there exist $B\to P$, and the square $A\to P \to Y$ // $A\to X \to Y$ is a pullback, so we have the following diagram in which the bottom and the whole squares are pullback, so is the upper square. If the category is Sets, $X\to Y$ is surjective then $A\to P $ is also surjective. Since the pullback of $B\to P$ along a surjective map is an bijection, $B\to P$ must be a bijection. This shows the right square of the original diagram is a pullback.

Suppose whole square and the left square in the diagram below are pullbacks, then we may wonder whether the right square is a pullback. It usually not the case.

Now we seek some addition condition on $X\to Y$ that forces the right square is a pullback too.

My question: is epic a sufficient condition? (If the category is Sets, then yes.)

Suppose whole square and the left square in the diagram below are pullbacks, then we may wonder whether the right square is a pullback. It is usually not the case.

Now we seek some addition condition on $X\to Y$ that forces the right square is a pullback too.

My question: is epic a sufficient condition? (If the category is Sets, then yes.)