FastCGI config in purely systemd: This works for services expecting a socket on stdin, like HG Web (see example)

fastcgi-server.service

[Unit]
Wants=network.service nginx.service

[Install]
Alias=hgweb
WantedBy=multi-user.target
[Service]
User=lbo
Type=exec
ExecStart=/home/lbo/hg/.config/hgweb.fcgi
StandardInput=socket
ExecStop=/usr/bin/kill $MAINPID
Restart=always

fastcgi-server.socket

[Unit]
Description = FCGI socket for hgweb

[Socket]
ListenStream = /run/web/hgweb.socket
Accept = false

[Install]
WantedBy = sockets.target hgweb.service

hgweb.fcgi

#!/usr/bin/env python2
#
# An example FastCGI script for use with flup, edit as necessary

# Path to repo or hgweb config to serve (see 'hg help hgweb')
config = "/home/lbo/hg/.config/hgweb_config"

# Uncomment and adjust if Mercurial is not installed system-wide
# (consult "installed modules" path from 'hg debuginstall'):
#import sys; sys.path.insert(0, "/path/to/python/lib")

# Uncomment to send python tracebacks to the browser if an error occurs:
#import cgitb; cgitb.enable()

from mercurial import demandimport; demandimport.enable()
from mercurial.hgweb import hgweb
from flup.server.fcgi import WSGIServer
application = hgweb(config)
WSGIServer(application).run()

nginx.conf

[...]
    location /lbo/hg/ {
        fastcgi_pass    unix:/run/web/hgweb.socket;
        include         fastcgi_hg;
        fastcgi_intercept_errors    on;

        limit_except GET HEAD {
          deny all;
        }
    }
[...]