Based on separate statements from both Sony and Microsoft, it has been confirmed that you will be able to connect the PlayStation 4 to the Xbox One's HDMI port and stream the PS4 through the Xbox One. Such as thing was possible in theory, but both companies had to confirm there were no blocking functions on either end. They did so without referring to the other.
Shuhei Yoshida confirmed on Twitter that they had decided to reverse policies so that the PlayStation 4 will not block any external device taking its HDMI feed to stream or record it. The big news was this will allow players to upload their play sessions to YouTube, but it can be used in other ways.
Microsoft, meanwhile, at their TGS presentation explained that the Xbox One would be able to take any video feed coming in from the HDMI port and stream it through the console. Later Albert Penello further detailed, stating that if consumers wanted to they could play Ryse and Killzone at the same time. Killzone is a PlayStation exclusive, so it is a rather roundabout way to saying yes your PS4 will stream through your Xbox One.
Not many people will do this, but it is nice for the consumers to have the options. Especially if they want an all in one media hub without having to change inputs or connections.