coshape, coequalizer, pushout (fiber coproduct), binary coproduct can be defined by reversing all arrows in the definitions of shape, equalizer, pullback (fiber product), binary product respectively.
The pushout equivalent of the fiber product object in pullback is the fiber coproduct object, denoted \(X +_{O} Y\), and the pushout equivalent of projections in pullback are injections, denoted \(i_X\) and \(i_Y\), respectively. The unique arrow of a pushout is denoted \([f, g]\).
The binary coproduct equivalent of the binary product object in binary product is the binary coproduct object, denoted \(X + Y\), and the binary coproduct equivalent of projections in binary product are injections, denoted \(i_X\) and \(i_Y\), respectively. The unique arrow of a binary coproduct is denoted \([f, g]\).
Diagramatically,
- Coshapes:
- coequalizer:
- pushout (fiber coproduct):
- binary coproduct:
given that the right-hand square is a pullback, the left-hand square is a pullback if and only if the outer rectangle is a pullback.