>I've an application which performs a 2 step process. It first use FTPGet to retreive the file from Unix-Box1 to a local destination. The second procedure use FTPPut to send the local file to Unix-Box2. What I would like to know, is there a way to actually 'get the file from Unix-Box1 and send to Unix-Box2' in one process (cutting out the middle step)?

Looking at this generally, what you have now is:
Unix-Box1: FTP Server
Unix-Box2: FTP Server
Your Local Machine: FTP client to both of the above servers

What you want is:
Unix-Box1: FTP Client
Unix-Box2: FTP Server

To do this ad-hoc (i.e. at your command, instigated from your machine), the admin of Unix-Box1 probably has to let you log in (securely, say via OpenSSH), and give you a command shell with enough rights to access its FTP client program and create/run a command or script, and to access the file you want to send to the other box. This is a lot more privileges than simple access to an FTP folder, and the admin may be reluctant to do this on security grounds.

On the other hand, in many cases file transfers like this don't need to be ad-hoc, they're often a scheduled task e.g. once every evening. In that case the Unix-Box1 admin could just set up a script and schedule it to be run regularly. You wouldn't have to do anything at all, and the admin wouldn't have to give you a remote shell.

One thing that may be a deal-breaker is that, whether you want direct transfers ad-hoc or scheduled, Unix-Box1 must know the credentials required (server name, user name, password, encryption key values(s), ...) to upload a file to Unix-Box2. In your current situation, Unix-Box1 doesn't need to know anything about Unix-Box2. This can be very desirable from a security standpoint. If Unix-Box2 doesn't trust Unix-Box1, this can be a real stumbling block.