Pentopia: One-way Crossing

_{This is an entry for Fortnightly Topic Challenge #44: Introduce a new grid deduction genre to the community.}

Here is a standard Pentopia puzzle.

Rules of Pentopia:

1. Place some pentominoes into the grid (not necessarily all of them) so that they do not touch each other, not even diagonally.
2. Pentominoes cannot repeat in the grid; rotations and reflections of a pentomino are considered the same shape.
3. The arrow denotes all of the 4 directions where the nearest pentominoes are located when looking from that square.
4. The arrow cannot be covered by any pentomino.