Likewise in German. Actually, more logical than English because we cannot say 'that morning' to refer to some other morning as it would be so indeterminate without further explanation.
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There is indeed a future tense in Hungarian, I wonder how these myths spread. It's like saying English has no future tense since present tenses can refer to future events under certain circumstances.
Because translation does not happen word by word, it needs to accomodate language specific rules. So even though hungarian does not need "a" for the morning, english does use it. I am not native english speaker, but I do not think "in morning" would be valid here.