The ornamental design for a baseball catching and throwingsystem, as shown and described.
FIG. 1 is a perspective view of a baseball catching and throwing system showing our new design;
FIG. 2 is a front view thereof;
FIG. 3 is a back view thereof;
FIG. 4 is a right side view thereof;
FIG. 5 is a left side view thereof;
FIG. 6 is a top view thereof; and,
FIG. 7 is a bottom view thereof.
The broken lines shown in FIGS. 1 and 2 are included for the purpose of illustrating portions of the baseball catching and throwing system that form no part of the claimed design.