How antagonistic muscle pairs work at the elbow during the downwards and upwards phase of a press up:
In the upward phase, the tricep contracts to extend the elbow, acting as the agonist, while the tricep relaxes and lengthens as the antagonist.
In the downwards phase, the bicep contracts eccentrically to control the downward movement, acting as the agonist, while the tricep relaxes and lengthens to allow elbow flexion.