Suppose I have a diagram $B \longleftarrow A \longrightarrow C$ in a category, and I execute a push-out operation and get $B \longrightarrow D \longleftarrow C$.
If I execute a pull-back over $B \longrightarrow D \longleftarrow C$, will I reach the same diagram $B \longleftarrow A \longrightarrow C$? Why?
As another question, suppose I execute a product operation over $A$ and $B$, and get $A \longleftarrow P \longrightarrow B$, also I execute co-product operation over the same $A$ and $B$, and get $A \longrightarrow CP \longleftarrow B$.
Is $A \longleftarrow P \longrightarrow B$ pull-back of $A \longrightarrow CP \longleftarrow B$?
How can I study the relations between limits and co-limits in general? Is there any reference discussing this in a consolidated manner.