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what was right in the eyes of Yahweh
Here Yahweh's "sight" refers to how he judges or decides on the value of something. See how you translated a similar phrase in 2 Chronicles 14:2. Alternate translation: "the things that Yahweh said were right" or "the things that Yahweh considers right" (See: rc://en/ta/man/jit/figs-metaphor)
walked in the ways of David his ancestor
Josiah behaving as David did is spoken of as if he walked on the same road or way as David. Alternate translation: "lived the way David his ancestor had lived" or "followed the example of David his ancestor" (See: rc://en/ta/man/jit/figs-metaphor)
did not turn away either to the right or to the left
To fully obey Yahweh is spoken of as if a person were on the correct road and never turned from it. Alternate translation: "did not do anything that would displease Yahweh" or "fully obeyed the laws of Yahweh" (See: rc://en/ta/man/jit/figs-metaphor)