Green’s Dictionary of Slang

cony-catcher n.2

also coney-catcher, conny-, cunny-
[cony n. (2)/cunny n. (1); play on cony-catcher n.1 ]

1. [late 16C–early 17C] a prostitute; thus cony-catching, prostitution.

2. [17C] the penis.

3. [17C–18C] a prostitute’s customer.