SSH from A through B to C, using private key on B [closed]

Schematic:

    ssh       ssh
A ------> B ------> C
    ^          ^
 using A's   using B's
 ssh key     ssh key

Preconditions:

• A is running ssh-agent;
• A can access B;
• B can access C;
• A‘s ssh public key is present in B:~/.ssh/authorized_keys
• B‘s ssh public key is present in C:~/.ssh/authorized_keys

In ~/.ssh/config on A, add

Host C
    ProxyCommand ssh -o 'ForwardAgent yes' B 'ssh-add && nc %h %p'

If your ssh private key on B is in a nonstandard location, add its path after ssh-add.

You should now be able to access C from A:

A$ ssh C
C$